Chen Chaotic System Research Articles

Constructive proof of Lagrange stability and sufficient – Necessary conditions of Lyapunov stability for Yang–Chen chaotic system

This paper studies the stability problem of Yang–Chen system. By introducing different radial unbounded Lyapunov functions in different regions, global exponential attractive set of Yang–Chen chaotic system is constructed with geometrical and algebraic methods. Then, simple algebraic sufficient and necessary conditions of global exponential stability, global asymptotic stability, and exponential instability of equilibrium are proposed. And the relevant expression of corresponding parameters for local exponential stability, local asymptotic stability, exponential instability of equilibria are obtained. Furthermore, the branch problem of the system is discussed, some branch expressions are given for the parameters of the system.

Identification of fractional-order systems with unknown initial values and structure

In this paper, the identification problem of fractional-order chaotic systems is proposed and investigated via an evolutionary optimization approach. Different with other studies to date, this research focuses on the identification of fractional-order chaotic systems with not only unknown orders and parameters, but also unknown initial values and structure. A group of fractional-order chaotic systems, i.e., Lorenz, Lü, Chen, Rössler, Arneodo and Volta chaotic systems, are set as the system candidate pool. The identification problem of fractional-order chaotic systems in this research belongs to mixed integer nonlinear optimization in essence. A powerful evolutionary algorithm called composite differential evolution (CoDE) is introduced for the identification problem presented in this paper. Extensive experiments are carried out to show that the fractional-order chaotic systems with unknown initial values and structure can be successfully identified by means of CoDE.

Mechanical Analysis and Energy Cycle of Chen Chaotic System

This paper formulates the Chen chaotic system to the extended Kolmogorov system, thereby investigating the physical meaning and energy cycle of Chen system. The Casimir function is introduced to analyze the system dynamics, and its time derivation is chosen to formulate energy cycle. The bound of chaotic attractor is obtained by the Casimir function and Lagrange multiplier. The Casimir function reflects the energy conversion and the distance between the orbit and the equilibria. These relationships are illustrated by numerical simulations.

Mechanical analysis of Chen chaotic system

The Chen chaotic system is transformed into Kolmogorov type system, which is decomposed into four types of torques: inertial torque, internal torque, dissipation and external torque. By the combinations of different torques, five cases are studied to discover key factors of chaos generation and the physical meaning. The conversion among Hamiltonian energy, kinetic energy and potential energy is investigated in these five cases. The relationship between the energies and the parameters is studied. It concludes that the combination of these four types of torques is necessary conditions to produce chaos, and any combination of three types of torques cannot produce chaos in Chen system.

A stack-based chaotic algorithm for encryption of colored images

In this paper, a new method is presented for encryption of colored images. This method is based on using stack data structure and chaos which make the image encryption algorithm more efficient and robust. In the proposed algorithm, a series of data whose range is between 0 and 3 is generated using chaotic logistic system. Then, the original image is divided into four subimages, and these four images are respectively pushed into the stack based on next number in the series. In the next step, the first element of the stack (which includes one of the four sub-images) is popped, and this image is divided into four other parts. Then, based on the next number in the series, four sub-images are pushed into the stack again. This procedure is repeated until the stack is empty. Therefore, during this process, each pixel unit is encrypted using another series of chaotic numbers (generated by Chen chaotic system). This method is repeated until all pixels of the plain image are encrypted. Finally, several extensive simulations on well-known USC datasets have been conducted to show the efficiency of this encryption algorithm. The tests performed showthat the proposed method has a really large key space and possesses high-entropic distribution. Consequently, it outperforms the other competing algorithms in the case of security

Counteracting Dynamical Degradation of Digital Chaotic Chebyshev Map via Perturbation

The chaotic map has complex dynamics under ideal conditions however it suffers from the problem of performance degradation in the case of finite computing precision. In order to prevent the dynamics degradation, in this paper the continuous Chen chaotic system is used to perturb both the inputs and parameters of Chebyshev map to minimize the chaotic degradation phenomenon under finite precision. Experimental evaluations and corresponding performance analysis demonstrate that the Chebyshev chaotic map has a good randomness and complex dynamic performance by using the proposed perturbation method, and some attributes of the proposed system are stronger than the original system (e.g. chaos attractor and approximate entropy). Finally, the corresponding pseudorandom number generator (PRNG) is constructed by this method and then its randomness is evaluated via NIST SP800-22 and TestU01 test suites, respectively. Statistical test results show that the proposed PRNG has high reliability of randomness, thus it can be used for cryptography and other potential applications.

Cryptanalysis of a Chaotic Key Based Image Encryption Scheme

Security of multimedia data is a major concern due to its widespread transmission over various communication channels. Hence design and study of good image encryption schemes has become a major research topic. During the last few decades, there has been a increasing in chaos-based cryptography. This paper proposes an attack on a recently proposed chaos based image encryption scheme. The cryptosystem under study proceed by first shuffling the original image to disturb the arrangement of pixels by applying a chaotic map several times. Second, a keystream is generated using Chen's chaotic system to mix it with the shuffled pixels to finally obtain the cipher image. A chosen ciphertext attack can be done to recover the system without any knowledge of the key. It simply demands two pairs of plaintext-ciphertext to completely break the cryptosystem.

Image Watermarking Encryption Scheme Based on Fractional Order Chaotic System

Abstract Now the chaotic system and wavelet transform are more and more widely used in the watermarking technology. At the same time, the fractional order chaotic system has more complex dynamic characteristics than the integer order system. So a new image watermarking scheme based on the fractional order Chen chaotic system and discrete wavelet transform is proposed. Chaotic sequences generated by chaotic system are used to encrypt the watermark image, and the processed watermark information is embedded into the original image by the discrete wavelet transform. Finally, the security analysis of the proposed watermarking algorithm is presented. The experimental results show that the proposed watermarking scheme has high security, and it has stronger robustness and invisibility compared with the previous work.

Synchronization of uncertain fractional-order chaotic systems with time delay based on adaptive neural network control

Time delay frequently appears in many phenomena of real life and the presence of time delay in a chaotic system leads to its complexity. It is of great practical significance to study the synchronization control of fractional-order chaotic systems with time delay. This is because it is closer to the real life and its dynamical behavior is more complex. However, the chaotic system is usually uncertain or unknown, and may also be affected by external disturbances, which cannot make the ideal model accurately describe the actual system. Moreover, in most of existing researches, they are difficult to realize the synchronization control of fractional-order time delay chaotic systems with unknown terms. In this paper, for the synchronization problems of the different structural fractional-order time delay chaotic systems with completely unknown nonlinear uncertain terms and external disturbances, based on Lyapunov stability theory, an adaptive radial basis function (RBF) neural network controller, which is accompanied by integer-order adaptive laws of parameters, is established. The controller combines RBF neural network and adaptive control technology, the RBF neural network is employed to approximate the unknown nonlinear functions, and the adaptive laws are used to adjust corresponding parameters of the controller. The system stability is analyzed by constructing a quadratic Lyapunov function. This method not only avoids the fractional derivative of the quadratic Lyapunov function, but also ensures that the adaptive laws are integer-order. Based on Barbalat lemma, it is proved that the synchronization error tends to zero asymptotically. In the numerical simulation, the uncertain fractional-order Liu chaotic system with time delay is chosen as the driving system, and the uncertain fractional-order Chen chaotic system with time delay is used as the response system. The simulation results show that the controller can realize the synchronization control of the different structural fractional-order chaotic systems with time delay, and has the advantages of fast response speed, good control effect, and strong anti-interference ability. From the perspective of long-term application, the synchronization of different structures has greater research significance and more development prospect than self synchronization. Therefore, the results of this study have great theoretical significance, and have a great application value in the field of secure communication.

A new S-box construction method based on the fractional-order chaotic Chen system

Since substitution box (S-box) is the only nonlinear component related to confusion properties for many block encryption algorithms, it is a necessity for the strong block encryption algorithms. S-box is a vital component in cryptography due to having the effect on the security of entire system. Therefore, alternative S-box construction techniques have been proposed in many researches. In this study, a new S-box construction method based on fractional-order (FO) chaotic Chen system is presented. In order to achieve that goal, numerical results of the FO chaotic Chen system for $$a= 35, b=3, c=28$$ and $$\alpha =0.9$$ are obtained by employing the predictor–corrector scheme. Besides, a simpler algorithm is suggested for the construction of S-box via time response of the FO chaotic Chen system. The performance of suggested S-box design is compared with other S-box designs developed by chaotic systems, and it is observed that this method provides a stronger S-box design.

Adaptive synchronization of two time-delayed fractional-order chaotic systems with different structure and different order

In this paper, the chaos synchronization of time-delayed fractional-order (FO) Lorenz chaotic systems and time-delayed FO Chen chaotic systems with different fractional derivative order is considered. First, by use of the fractional calculus techniques, we give the phase trajectories of the time-delayed FO Lorenz and time-delayed FO Chen chaotic systems. Then, combined the active control and adaptive control theory, a novel synchronization scheme, which includes the compensation controller and the optimal controller, is proposed for two time-delayed FO chaotic systems with different structure and different order in presence of parameter uncertainty. Based on the new version of FO Lyapunov stability theory, we design the compensation controllers and optimal controllers, respectively. Finally, numerical simulations have shown to illustrate the effectiveness and validation of the proposed method.

Different Generalized Synchronization Schemes Between Integer-Order and Fractional-Order Chaotic Systems with Different Dimensions

This paper addresses the problem of generalized synchronization (GS) between different dimensional integer-order and fractional-order chaotic systems. Based on the stability theory of linear continuous time dynamical systems, stability results of linear fractional order systems and nonlinear controllers, different criterions are derived to achieve generalized synchronization. The effectiveness of the proposed control schemes are verified by considering two examples: fractional-order chaotic Lorenz and hyperchaotic Lorenz systems and hyperchaotic Chen and fractional-order chaotic Chen systems.

A Class of Integer Order and Fractional Order Hyperchaotic Systems via the Chen System

In this article, we investigate the generation of a class of hyperchaotic systems via the Chen chaotic system using both integer order and fractional order differential equation systems. Based on the Chen chaotic system, we designed a system with four nonlinear ordinary differential equations. For different parameter sets, the trajectory of the system may diverge or display a hyperchaotic attractor with double wings. By linearizing the ordinary differential equation system with divergent trajectory and designing proper switching controls, we obtain a chaotic attractor. Similar phenomenon has also been observed in linearizing the hyperchaotic system. The corresponding fractional order systems are also considered. Our investigation indicates that, switching control can be applied to either linearized chaotic or nonchaotic differential equation systems to create chaotic attractor.

Adaptive generalized combination complex synchronization of uncertain real and complex nonlinear systems

With comprehensive consideration of generalized synchronization, combination synchronization and adaptive control, this paper investigates a novel adaptive generalized combination complex synchronization (AGCCS) scheme for different real and complex nonlinear systems with unknown parameters. On the basis of Lyapunov stability theory and adaptive control, an AGCCS controller and parameter update laws are derived to achieve synchronization and parameter identification of two real drive systems and a complex response system, as well as two complex drive systems and a real response system. Two simulation examples, namely, ACGCS for chaotic real Lorenz and Chen systems driving a hyperchaotic complex Lü system, and hyperchaotic complex Lorenz and Chen systems driving a real chaotic Lü system, are presented to verify the feasibility and effectiveness of the proposed scheme.

Complex Generalized Synchronization and Parameter Identification of Nonidentical Nonlinear Complex Systems.

In this paper, generalized synchronization (GS) is extended from real space to complex space, resulting in a new synchronization scheme, complex generalized synchronization (CGS). Based on Lyapunov stability theory, an adaptive controller and parameter update laws are designed to realize CGS and parameter identification of two nonidentical chaotic (hyperchaotic) complex systems with respect to a given complex map vector. This scheme is applied to synchronize a memristor-based hyperchaotic complex Lü system and a memristor-based chaotic complex Lorenz system, a chaotic complex Chen system and a memristor-based chaotic complex Lorenz system, as well as a memristor-based hyperchaotic complex Lü system and a chaotic complex Lü system with fully unknown parameters. The corresponding numerical simulations illustrate the feasibility and effectiveness of the proposed scheme.

Unknown input nonlinear observer design for continuous and discrete time systems with input recovery scheme

In the present work, nonlinear observer design using differential mean value (DMV) theorem is extended to the design of nonlinear unknown input observer. To address the unknown input, famous approach of decoupling the unknown input has been used. Here, by the use of differential mean value theorem, the problem of designing unknown input observer evolves as stabilization of linear parameter varying system. On the basis of Lyapunov approach, linear matrix inequality (LMI) conditions for the existence of such observer are derived. For nonlinear systems, large value of Lipschitz constant may render LMI infeasible. DMV theorem may address the issue by replacing Lipschitz constants with more refined constants of lesser magnitude. DMV theorem-based approach is shown to be less conservative compared to Lipschitz constant. The proposed approach is shown to be effective for both continuous and discrete time systems. For discrete time systems, unknown input recovery scheme is also presented. Extensive simulations are shown in the end for continuous time chaotic Chen system, discrete Lorenz chaotic system and flexible joint robot link system, to highlight the efficacy of proposed strategy.

Fast synchronization of non-identical chaotic modulation-based secure systems using a modified sliding mode controller

In this paper, a secure communication scheme based on chaotic modulation is proposed using a reversible process and a robust controller with efficient cost and complexity to synchronize two different chaotic systems. In the controller design, a sliding mode control with an adaptive rule is used for non-linear inputs. The adaptive rule is applied to ensure the synchronization when uncertainties, non-modeled dynamics or external distortions are at work. The message signal is recovered at the receiver using a recursive process at the end. The effectiveness of the proposed algorithm is confirmed via the simulation results for the synchronization of the transmitted signal modulated by Chen chaotic system at the transmitter and Genesio chaotic system at the receiver, and those for the information recovery process.

An Improved Cuckoo Search Optimization Algorithm for the Problem of Chaotic Systems Parameter Estimation.

This paper proposes an improved cuckoo search (ICS) algorithm to establish the parameters of chaotic systems. In order to improve the optimization capability of the basic cuckoo search (CS) algorithm, the orthogonal design and simulated annealing operation are incorporated in the CS algorithm to enhance the exploitation search ability. Then the proposed algorithm is used to establish parameters of the Lorenz chaotic system and Chen chaotic system under the noiseless and noise condition, respectively. The numerical results demonstrate that the algorithm can estimate parameters with high accuracy and reliability. Finally, the results are compared with the CS algorithm, genetic algorithm, and particle swarm optimization algorithm, and the compared results demonstrate the method is energy-efficient and superior.

Robust Fractional Order Controller for Chaotic Systems

In this paper, we propose a robust fractional controller structure to control chaotic systems. The fractional controller structure is based on the predictive control of chaotic systems. The parameters of the controller are obtained by minimizing the error state energy.Simulation results on Chen and Lorenz chaotic systems show the effectiveness of the proposed controller to reject the noise and the disturbance injected to these systems states. These results show also that the proposed controller gives better results than the standard predictive control.

Compound Generalized Function Projective Synchronization for Fractional-Order Chaotic Systems

A modified function projective synchronization for fractional-order chaotic system, called compound generalized function projective synchronization (CGFPS), is proposed theoretically in this paper. There are one scaling-drive system, more than one base-drive system, and one response system in the scheme of CGFPS, and the scaling function matrices come from multidrive systems. The proposed CGFPS technique is based on the stability theory of fractional-order system. Moreover, we achieve the CGFPS between three-driver chaotic systems, that is, the fractional-order Arneodo chaotic system, the fractional-order Chen chaotic system, and the fractional-order Lu chaotic system, and one response chaotic system, that is, the fractional-order Lorenz chaotic system. Numerical experiments are demonstrated to verify the effectiveness of the CGFPS scheme.